Questions: Use the given right triangle to find ratios, in reduced form, for sin A, cos A, and tan A. Enter the ratios in reduced form: sin A=

Transcript text: Use the given right triangle to find ratios, in reduced form, for $\sin \mathrm{A}, \cos \mathrm{A}$, and $\tan \mathrm{A}$. Enter the ratios in reduced form: \[ \sin A= \]

Solution

Solution Steps

Step 1: Find the length of side BC

We are given a right triangle ABC with angle C being the right angle. We are given the lengths of sides AC and AB as 48 and 50 respectively. We can use the Pythagorean theorem to find the length of side BC. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In our case, AB is the hypotenuse, so we have:

$AB^2 = AC^2 + BC^2$ $50^2 = 48^2 + BC^2$ $2500 = 2304 + BC^2$ $BC^2 = 2500 - 2304$ $BC^2 = 196$ $BC = \sqrt{196}$ $BC = 14$

Step 2: Calculate sin A

The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In our triangle, the side opposite angle A is BC, and the hypotenuse is AB. Therefore:

$\sin A = \frac{BC}{AB} = \frac{14}{50} = \frac{7}{25}$

Step 3: Calculate cos A

The cosine of an angle in a right triangle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. In our triangle, the side adjacent to angle A is AC, and the hypotenuse is AB. Therefore:

$\cos A = \frac{AC}{AB} = \frac{48}{50} = \frac{24}{25}$

Step 4: Calculate tan A

The tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In our triangle, the side opposite angle A is BC, and the side adjacent to angle A is AC. Therefore:

$\tan A = \frac{BC}{AC} = \frac{14}{48} = \frac{7}{24}$